In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

Formal definition edit

Let   and   be measurable spaces, meaning that   and   are sets equipped with respective  -algebras   and   A function   is said to be measurable if for every   the pre-image of   under   is in  ; that is, for all  


That is,   where   is the σ-algebra generated by f. If   is a measurable function, one writes

to emphasize the dependency on the  -algebras   and  

Term usage variations edit

The choice of  -algebras in the definition above is sometimes implicit and left up to the context. For example, for     or other topological spaces, the Borel algebra (generated by all the open sets) is a common choice. Some authors define measurable functions as exclusively real-valued ones with respect to the Borel algebra.[1]

If the values of the function lie in an infinite-dimensional vector space, other non-equivalent definitions of measurability, such as weak measurability and Bochner measurability, exist.

Notable classes of measurable functions edit

  • Random variables are by definition measurable functions defined on probability spaces.
  • If   and   are Borel spaces, a measurable function   is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see Luzin's theorem. If a Borel function happens to be a section of a map   it is called a Borel section.
  • A Lebesgue measurable function is a measurable function   where   is the  -algebra of Lebesgue measurable sets, and   is the Borel algebra on the complex numbers   Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. In the case     is Lebesgue measurable if and only if   is measurable for all   This is also equivalent to any of   being measurable for all   or the preimage of any open set being measurable. Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable.[2] A function   is measurable if and only if the real and imaginary parts are measurable.

Properties of measurable functions edit

  • The sum and product of two complex-valued measurable functions are measurable.[3] So is the quotient, so long as there is no division by zero.[1]
  • If   and   are measurable functions, then so is their composition  [1]
  • If   and   are measurable functions, their composition   need not be  -measurable unless   Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.
  • The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.[1][4]
  • The pointwise limit of a sequence of measurable functions   is measurable, where   is a metric space (endowed with the Borel algebra). This is not true in general if   is non-metrizable. The corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.[5][6]

Non-measurable functions edit

Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions. Such proofs rely on the axiom of choice in an essential way, in the sense that Zermelo–Fraenkel set theory without the axiom of choice does not prove the existence of such functions.

In any measure space   with a non-measurable set     one can construct a non-measurable indicator function:

where   is equipped with the usual Borel algebra. This is a non-measurable function since the preimage of the measurable set   is the non-measurable    

As another example, any non-constant function   is non-measurable with respect to the trivial  -algebra   since the preimage of any point in the range is some proper, nonempty subset of   which is not an element of the trivial  

See also edit

Notes edit

  1. ^ a b c d Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6.
  2. ^ Carothers, N. L. (2000). Real Analysis. Cambridge University Press. ISBN 0-521-49756-6.
  3. ^ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0-471-31716-0.
  4. ^ Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0-02-404151-3.
  5. ^ Dudley, R. M. (2002). Real Analysis and Probability (2 ed.). Cambridge University Press. ISBN 0-521-00754-2.
  6. ^ Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis, A Hitchhiker's Guide (3 ed.). Springer. ISBN 978-3-540-29587-7.

External links edit